Systems and methods for real-time estimation of capacity of a rechargeable battery

ABSTRACT

A battery system includes a battery that couples to an electrical system. The battery system also includes a battery control module that electrically couples to the battery. The battery control module monitors at least one monitored parameter of the battery, and the battery control module recursively calculates a first capacity estimation of the battery using two linear regression models based on at least an equivalent circuit model, the at least one monitored parameter, and a Kalman filter.

BACKGROUND

The present disclosure generally relates to the field of batteries andbattery modules. More specifically, the present disclosure relates toestimating real-time capacity of a rechargeable battery.

This section is intended to introduce the reader to various aspects ofart that may be related to various aspects of the present disclosure,which are described below. This discussion is believed to be helpful inproviding the reader with background information to facilitate a betterunderstanding of the various aspects of the present disclosure.Accordingly, it should be understood that these statements are to beread in this light, and not as admissions of prior art.

A vehicle that uses one or more battery systems for providing all or aportion of the motive power for the vehicle can be referred to as anxEV, where the term “xEV” is defined herein to include all of thefollowing vehicles, or any variations or combinations thereof, that useelectric power for all or a portion of their vehicular motive force. Forexample, xEVs include electric vehicles (EVs) that utilize electricpower for all motive force. As will be appreciated by those skilled inthe art, hybrid electric vehicles (HEVs), also considered xEVs, combinean internal combustion engine propulsion system and a battery-poweredelectric propulsion system, such as 48 Volt (V) or 130V systems. Theterm HEV may include any variation of a hybrid electric vehicle. Forexample, full hybrid systems (FHEVs) may provide motive and otherelectrical power to the vehicle using one or more electric motors, usingonly an internal combustion engine, or using both. In contrast, mildhybrid systems (MHEVs) disable the internal combustion engine when thevehicle is idling and utilize a battery system to continue powering theair conditioning unit, radio, or other electronics, as well as torestart the engine when propulsion is desired. The mild hybrid systemmay also apply some level of power assist, during acceleration forexample, to supplement the internal combustion engine. Mild hybrids aretypically 96V to 130V and recover braking energy through a belt or crankintegrated starter generator. Further, a micro-hybrid electric vehicle(mHEV) also uses a “Stop-Start” system similar to the mild hybrids, butthe micro-hybrid systems of a mHEV may or may not supply power assist tothe internal combustion engine and operates at a voltage below 60V. Forthe purposes of the present discussion, it should be noted that mHEVstypically do not technically use electric power provided directly to thecrankshaft or transmission for any portion of the motive force of thevehicle, but an mHEV may still be considered as an xEV since it does useelectric power to supplement a vehicle's power needs when the vehicle isidling with internal combustion engine disabled and recovers brakingenergy through an integrated starter generator. In addition, a plug-inelectric vehicle (PEV) is any vehicle that can be charged from anexternal source of electricity, such as wall sockets, and the energystored in the rechargeable battery packs drives or contributes to drivethe wheels. PEVs are a subcategory of EVs that include all-electric orbattery electric vehicles (BEVs), plug-in hybrid electric vehicles(PHEVs), and electric vehicle conversions of hybrid electric vehiclesand conventional internal combustion engine vehicles.

xEVs as described above may provide a number of advantages as comparedto more traditional gas-powered vehicles using only internal combustionengines and traditional electrical systems, which are typically 12Vsystems powered by a lead acid battery. For example, xEVs may producefewer undesirable emission products and may exhibit greater fuelefficiency as compared to traditional internal combustion vehicles and,in some cases, such xEVs may eliminate the use of gasoline entirely, asis the case of certain types of EVs or PEVs.

As technology continues to evolve, there is a need to provide improvedstate indicators for battery modules of such vehicles. For example, theelectric power used by the xEVs may be provided by rechargeablebatteries. It may be difficult to accurately depict a state of charge orcapacity of the rechargeable batteries while the rechargeable batteriesare in operation. The present disclosure is generally related toestimating real-time parameters of the rechargeable battery duringoperation of the rechargeable battery and/or the xEV.

SUMMARY

A summary of certain embodiments disclosed herein is set forth below. Itshould be understood that these aspects are presented merely to providethe reader with a brief summary of these certain embodiments and thatthese aspects are not intended to limit the scope of this disclosure.Indeed, this disclosure may encompass a variety of aspects that may notbe set forth below.

The present disclosure relates to a battery system that includes abattery that couples to an electrical system. The battery system alsoincludes a battery control module that electrically couples to thebattery. The battery control module monitors at least one monitoredparameter of the battery, and the battery control module recursivelycalculates a first capacity estimation of the battery using two linearregression models. The two linear regression models are based on atleast an equivalent circuit model, the at least one monitored parameter,and a Kalman filter.

The present disclosure also relates to a method to determine real-timecapacity of a rechargeable battery that couples to an electrical system.The method includes monitoring at least one monitored parameter of therechargeable battery during operation of the battery with a batterycontrol module via one or more sensors coupled to the rechargeablebattery. The method also includes recursively calculating a firstreal-time capacity of the rechargeable battery via the battery controlmodule using two linear regression models. The two linear regressionmodels are based on at least an equivalent circuit model, the at leastone monitored parameter, and a Kalman filter.

The present disclosure also relates to an energy storage component foruse in a vehicle. The energy storage component includes a housing, afirst terminal and a second terminal, and a rechargeable batterydisposed in the housing. The rechargeable battery couples to the firstterminal and the second terminal. The energy storage component alsoincludes a battery control module that monitors at least one monitoredparameter of the energy storage component. The battery control modulealso recursively calculates a first capacity estimation of the energystorage component using two linear regression models based on at leastan equivalent circuit model, the at least one monitored parameter, and aKalman filter. Additionally, the battery control module recursivelycalculates a second capacity estimation of the energy storage componentbased at least in part on two relaxation open circuit voltagemeasurements of the energy storage component and coulomb counting of theenergy storage component

DRAWINGS

Various aspects of this disclosure may be better understood upon readingthe following detailed description and upon reference to the drawings inwhich:

FIG. 1 is perspective view of a vehicle (an xEV) having a battery systemcontributing all or a portion of the power for the vehicle, inaccordance with an embodiment of the present approach.

FIG. 2 is a cutaway schematic view of the xEV of FIG. 1 in the form of ahybrid electric vehicle (HEV), in accordance with an embodiment of thepresent approach;

FIG. 3 is a schematic view of a battery system of the xEV of FIG. 1, inaccordance with an embodiment of the present approach;

FIG. 4 is a 1-RC equivalent circuit model of an energy storage componentof the xEV of FIG. 1, in accordance with an embodiment of the presentapproach;

FIG. 5 is a chart of a relationship between an open circuit voltage(OCV) and a state of charge (SOC) of an energy storage component of thexEV of FIG. 1, in accordance with an embodiment of the present approach;

FIGS. 6A and 6B are a process flow diagram describing a method forcalculating energy storage component parameters and determiningparameter convergence of the energy storage component, in accordancewith an embodiment of the present approach;

FIGS. 7A and 7B are ring buffers for storing data calculated using theprocess flow diagram of FIGS. 6A and 6B, in accordance with anembodiment of the present approach;

FIGS. 8A and 8B are a process flow diagram describing a method forcalculating a capacity of the energy storage component using two linearregression models, in accordance with an embodiment of the presentapproach;

FIGS. 9A and 9B are a process flow diagram describing a method forcalculating a capacity of the energy storage component using tworelaxation open circuit voltage measurements and coulomb counting, inaccordance with an embodiment of the present approach;

FIG. 10 is a process flow diagram describing a method for directionallyvalidating a capacity estimation of the energy storage component, inaccordance with an embodiment; and

FIG. 11 is a chart depicting an embodiment of the process flow diagramof FIG. 10, in accordance with an embodiment.

DETAILED DESCRIPTION

One or more specific embodiments will be described below. In an effortto provide a concise description of these embodiments, not all featuresof an actual implementation are described in the specification. Itshould be appreciated that in the development of any such actualimplementation, as in any engineering or design project, numerousimplementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness-related constraints, which may vary from one implementation toanother. Moreover, it should be appreciated that such a developmenteffort might be complex and time consuming, but would nevertheless be aroutine undertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure.

The battery systems described herein may be used to provide power tovarious types of electric vehicles (xEVs) and other high voltage energystorage/expending applications (e.g., electrical grid power storagesystems). Such battery systems may include one or more battery modules,each battery module having a number of battery cells (e.g., lithium-ion(Li-ion) electrochemical cells) arranged and electrically interconnectedto provide particular voltages and/or currents useful to power, forexample, one or more components of an xEV. As another example, batterymodules in accordance with present embodiments may be incorporated withor provide power to stationary power systems (e.g., non-automotivesystems).

Based on the advantages over traditional gas-power vehicles,manufactures, which generally produce traditional gas-powered vehicles,may desire to utilize improved vehicle technologies (e.g., regenerativebraking technology) within their vehicle lines. Often, thesemanufacturers may utilize one of their traditional vehicle platforms asa starting point. Accordingly, since traditional gas-powered vehiclesare designed to utilize 12 volt battery systems, a 12 volt lithium ionbattery may be used to supplement a 12 volt lead-acid battery. Morespecifically, the 12 volt lithium ion battery may be used to moreefficiently capture electrical energy generated during regenerativebraking and subsequently supply electrical energy to power the vehicle'selectrical system.

As advancements occur with vehicle technologies, high voltage electricaldevices may also be included in the vehicle's electrical system. Forexample, the lithium ion battery may supply electrical energy to anelectric motor in a mild-hybrid vehicle. Often, these high voltageelectrical devices utilize voltage greater than 12 volts, for example,up to 48 volts. Accordingly, in some embodiments, the output voltage ofa 12 volt lithium ion battery may be boosted using a DC-DC converter tosupply power to the high voltage devices. Additionally or alternatively,a 48 volt lithium ion battery may be used to supplement a 12 voltlead-acid battery. More specifically, the 48 volt lithium ion batterymay be used to more efficiently capture electrical energy generatedduring regenerative braking and subsequently supply electrical energy topower the high voltage devices.

Thus, the design choice regarding whether to utilize a 12 volt lithiumion battery or a 48 volt lithium ion battery may depend directly on theelectrical devices included in a particular vehicle. Nevertheless,although the voltage characteristics may differ, the operationalprinciples of a 12 volt lithium ion battery and a 48 volt lithium ionbattery are generally similar. More specifically, as described above,both may be used to capture electrical energy during regenerativebraking and subsequently supply electrical energy to power electricaldevices in the vehicle.

Accordingly, to simplify the following discussion, the presenttechniques will be described in relation to a battery system with a 12volt lithium ion battery and a 12 volt lead-acid battery. However, oneof ordinary skill in art is able to adapt the present techniques toother battery systems, such as a battery system with a 48 volt lithiumion battery and a 12 volt lead-acid battery.

The present disclosure relates to batteries and battery modules. Morespecifically, the present disclosure relates estimating real-timeparameters of rechargeable batteries. Particular embodiments aredirected to lithium ion battery cells that may be used in vehicularcontexts (e.g., hybrid electric vehicles) as well as other energystorage/expending applications (e.g., energy storage for an electricalgrid).

With the preceding in mind, the present disclosure describes techniquesfor estimating real-time parameters of the rechargeable batteries.Traditionally, a rated capacity of a rechargeable battery may bedetermined by completely discharging the rechargeable battery from afully charged state using a constant discharge rate at room temperature.Such a method may not be practical for rechargeable batteries used invehicular contexts. Accordingly, a partial discharge of the rechargeablebattery may be used to estimate a capacity of the rechargeable battery.Accordingly, a battery control unit described in the present disclosuremay estimate the capacity and other parameters of the rechargeablebattery in real-time or near real-time using the systems and methodsdescribed in detail below.

To help illustrate, FIG. 1 is a perspective view of an embodiment of avehicle 10, which may utilize a regenerative braking system. Althoughthe following discussion is presented in relation to vehicles withregenerative braking systems, the techniques described herein areadaptable to other vehicles that capture/store electrical energy with abattery, which may include electric-powered and gas-powered vehicles.

As discussed above, it would be desirable for a battery system 12 to belargely compatible with traditional vehicle designs. Accordingly, thebattery system 12 may be placed in a location in the vehicle 10 thatwould have housed a traditional battery system. For example, asillustrated, the vehicle 10 may include the battery system 12 positionedsimilarly to a lead-acid battery of a typical combustion-engine vehicle(e.g., under the hood of the vehicle 10). Furthermore, as will bedescribed in more detail below, the battery system 12 may be positionedto facilitate managing temperature of the battery system 12. Forexample, in some embodiments, positioning a battery system 12 under thehood of the vehicle 10 may enable an air duct to channel airflow overthe battery system 12 and cool the battery system 12.

A more detailed view of the battery system 12 is described in FIG. 2. Asdepicted, the battery system 12 includes an energy storage component 14coupled to an ignition system 16, an alternator 18, a vehicle console20, and optionally to an electric motor 22. Generally, the energystorage component 14 may capture/store electrical energy generated inthe vehicle 10 and output electrical energy to power electrical devicesin the vehicle 10.

In other words, the battery system 12 may supply power to components ofthe vehicle's electrical system, which may include radiator coolingfans, climate control systems, electric power steering systems, activesuspension systems, auto park systems, electric oil pumps, electricsuper/turbochargers, electric water pumps, heated windscreen/defrosters,window lift motors, vanity lights, tire pressure monitoring systems,sunroof motor controls, power seats, alarm systems, infotainmentsystems, navigation features, lane departure warning systems, electricparking brakes, external lights, or any combination thereof.Illustratively, in the depicted embodiment, the energy storage component14 supplies power to the vehicle console 20, a display 21 within thevehicle, and the ignition system 16, which may be used to start (e.g.,crank) an internal combustion engine 24.

Additionally, the energy storage component 14 may capture electricalenergy generated by the alternator 18 and/or the electric motor 22. Insome embodiments, the alternator 18 may generate electrical energy whilethe internal combustion engine 24 is running. More specifically, thealternator 18 may convert the mechanical energy produced by the rotationof the internal combustion engine 24 into electrical energy.Additionally or alternatively, when the vehicle 10 includes an electricmotor 22, the electric motor 22 may generate electrical energy byconverting mechanical energy produced by the movement of the vehicle 10(e.g., rotation of the wheels) into electrical energy. Thus, in someembodiments, the energy storage component 14 may capture electricalenergy generated by the alternator 18 and/or the electric motor 22during regenerative braking. As such, the alternator 18 and/or theelectric motor 22 are generally referred to herein as a regenerativebraking system.

To facilitate capturing and supplying electric energy, the energystorage component 14 may be electrically coupled to the vehicle'selectric system via a bus 26. For example, the bus 26 may enable theenergy storage component 14 to receive electrical energy generated bythe alternator 18 and/or the electric motor 22. Additionally, the bus 26may enable the energy storage component 14 to output electrical energyto the ignition system 16 and/or the vehicle console 20. Accordingly,when a 12 volt battery system 12 is used, the bus 26 may carryelectrical power typically between 8-18 volts.

Additionally, as depicted, the energy storage component 14 may includemultiple battery modules. For example, in the depicted embodiment, theenergy storage component 14 includes a lead acid (e.g., a first) batterymodule 28 in accordance with present embodiments, and a lithium ion(e.g., a second) battery module 30, where each battery module 28, 30includes one or more battery cells. In other embodiments, the energystorage component 14 may include any number of battery modules.Additionally, although the first battery module 28 and the secondbattery module 30 are depicted adjacent to one another, they may bepositioned in different areas around the vehicle. For example, thesecond battery module 30 may be positioned in or about the interior ofthe vehicle 10 while the first battery module 28 may be positioned underthe hood of the vehicle 10.

In some embodiments, the energy storage component 14 may includemultiple battery modules to utilize multiple different batterychemistries. For example, the first battery module 28 may utilize alead-acid battery chemistry and the second battery module 30 may utilizea lithium ion battery chemistry. In such an embodiment, the performanceof the battery system 12 may be improved since the lithium ion batterychemistry generally has a higher coulombic efficiency and/or a higherpower charge acceptance rate (e.g., higher maximum charge current orcharge voltage) than the lead-acid battery chemistry. As such, thecapture, storage, and/or distribution efficiency of the battery system12 may be improved.

To facilitate controlling the capturing and storing of electricalenergy, the battery system 12 may additionally include a control module32. More specifically, the control module 32 may control operations ofcomponents in the battery system 12, such as relays (e.g., switches)within energy storage component 14, the alternator 18, and/or theelectric motor 22. For example, the control module 32 may regulateamount of electrical energy captured/supplied by each battery module 28or 30 (e.g., to de-rate and re-rate the battery system 12), perform loadbalancing between the battery modules 28 and 30, determine a state ofcharge of each battery module 28 or 30, determine temperature of eachbattery module 28 or 30, determine a predicted temperature trajectory ofeither battery module 28 and 30, determine predicted life span of eitherbattery module 28 or 30, determine fuel economy contribution by eitherbattery module 28 or 30, control magnitude of voltage or current outputby the alternator 18 and/or the electric motor 22, and the like.

Accordingly, the control module (e.g., unit) 32 may include one or moreprocessors 34 and one or more memories 36. More specifically, the one ormore processors 34 may include one or more application specificintegrated circuits (ASICs), one or more field programmable gate arrays(FPGAs), one or more general purpose processors, or any combinationthereof. Generally, the processor 34 may perform computer-readableinstructions related to the processes described herein. Additionally,the processor 34 may be a fixed-point processor or a floating-pointprocessor.

Additionally, the one or more memories 36 may include volatile memory,such as random access memory (RAM), and/or non-volatile memory, such asread-only memory (ROM), optical drives, hard disc drives, or solid-statedrives. In some embodiments, the control module 32 may include portionsof a vehicle control unit (VCU) and/or a separate battery controlmodule. Additionally, as depicted, the control module 32 may be includedseparate from the energy storage component 14, such as a standalonemodule. In other embodiments, the battery management system 36 may beincluded within the energy storage component 14.

In certain embodiments, the control module 32 or the processor 34 mayreceive data from various sensors 38 disposed within and/or around theenergy storage component 14. The sensors 38 may include a variety ofsensors for measuring current, voltage, temperature, and the likeregarding the battery module 28 or 30. After receiving data from thesensors 38, the processor 34 may convert raw data into estimations ofparameters of the battery modules 28 and 30. As such, the processor 34may render the raw date into data that may provide an operator of thevehicle 10 with valuable information pertaining to operations of thebattery system 12, and the information pertaining to the operations ofthe battery system 12 may be displayed on the display 21. The display 21may display various images generated by device 10, such as a GUI for anoperating system or image data (including still images and video data).The display 21 may be any suitable type of display, such as a liquidcrystal display (LCD), plasma display, or an organic light emittingdiode (OLED) display, for example. Additionally, the display 21 mayinclude a touch-sensitive element that may provide inputs to the adjustparameters of the control module 32 or data processed by the processor34.

The energy storage component 14 may have dimensions comparable to thoseof a typical lead-acid battery to limit modifications to the vehicle 10design to accommodate the battery system 12. For example, the energystorage component 14 may be of similar dimensions to an H6 battery,which may be approximately 13.9 inches×6.8 inches×7.5 inches. Asdepicted, the energy storage component 14 may be included within asingle continuous housing. In other embodiments, the energy storagecomponent 14 may include multiple housings coupled together (e.g., afirst housing including the first battery 28 and a second housingincluding the second battery 30). In still other embodiments, asmentioned above, the energy storage component 14 may include the firstbattery module 28 located under the hood of the vehicle 10, and thesecond battery module 30 may be located within the interior of thevehicle 10.

More specifically, FIG. 3 illustrates a schematic view of components ofthe battery system 12. As mentioned above in the discussion of FIG. 2,the control module 32 may regulate amount of electrical energycaptured/supplied by each battery module 28 or 30 (e.g., to de-rate andre-rate the battery system 12), perform load balancing between thebattery modules 28 and 30, determine a state of charge of each batterymodule 28 or 30, determine temperature of each battery module 28 or 30,determine a predicted temperature trajectory of either battery module 28or 30, determine predicted life span of either battery module 28 or 30,determine fuel economy contribution by either battery module 28 or 30,control magnitude of voltage or current output by the alternator 18and/or the electric motor 22, and the like. In particular, the controlmodule 32 may enable measurement of the state of charge (SOC) and/orstate of health (SOH) based on battery parameters measured by thesensors 38.

In some embodiments, the energy storage component 14 may include asingle lithium ion cell or a plurality of lithium ion cells coupled inseries. Additionally, other rechargeable battery chemistries arecontemplated. The energy storage component 14 may discharge storedenergy to a load 40, which may include the ignition system 16, thevehicle console 20, the display 21, the electric motor 22, and any otherelectric components of the vehicle 10. As the energy storage component14 discharges the stored energy to the load 40, the alternator 18 and/orthe electric motor 22 may provide energy to the energy storage component14 to replenish the stored energy previously discharged to the load 40.The sensors 38 may measure battery parameters of the energy storagecomponent 14, and the sensors 38 may transmit the measurements to thecontrol module 32. The battery parameters of the energy storagecomponent 14 may include terminal voltage measurements, terminal currentmeasurements, and battery temperature measurements. The control module32 processes the measured battery parameters, as described in detailbelow, to estimate the SOC of the energy storage component 14, tworesistances associated with an equivalent circuit model of the energystorage component 14, and a capacitance associated with the equivalentcircuit model. As discussed below in relation to FIG. 4, the equivalentcircuit model may be a 1-RC equivalent circuit model.

Further, it may be appreciated that the systems and methods describedherein may be used for varying chemistries of the energy storagecomponent 14. For example, the SOC, the resistances, and the capacitanceof the energy storage component 14 may represent a single or multi-celllithium ion battery, a single or multi-cell lead-acid battery, somecombination thereof (e.g., a lithium ion battery electrically coupled inparallel to a lead-acid battery), or any other single or multi-cellbattery chemistries. Furthermore, in energy storage components 14 withmultiple battery chemistries electrically coupled in parallel, the SOC,the resistances, and the capacitance may represent the entire energystorage component 14, or the SOC, the resistances, and the capacitancemay be calculated for each of the multiple battery chemistries.

FIG. 4 depicts a 1-RC equivalent circuit model 42 of the energy storagecomponent 14. The 1-RC equivalent circuit model 42 relates batteryparameters (e.g., open circuit voltage (OCV) 44, resistances 46 and 48,and capacitance 50) to the measured parameters (e.g., terminal voltage52, terminal current, and battery temperature) measured by the sensors38. Additionally, the 1-RC equivalent circuit model 42 provides amechanism to estimate the OCV in real-time during operation of thevehicle 10. Using other methods to measure the OCV, such as a coulombcounting method, the OCV may be measurable when the battery system 12 isat a resting state for an extended amount of time. That is, the OCV ofthe battery system 12 may be measurable after the battery system 12 hasrested for one or more hours. Accordingly, by using the 1-RC equivalentcircuit model 42, a rest period is no longer used, and the OCV may beestimated while the battery system 12 operates under the load 40.

In the 1-RC equivalent circuit model 42, the resistance 46 (i.e., R₀)represents an ohmic resistance of a current path of the energy storagecomponent 14, the resistance 48 (e.g. R₁) represents a charge transferresistance of the energy storage component 14, and the capacitance 50(e.g. C₁) represents a double layer capacitance of the energy storagecomponent 14. The 1-RC equivalent circuit model 42 is referred to as a1-RC equivalent circuit model due to the single resistor-capacitorpairing (e.g., the resistance 48 and the capacitance 50). Using the 1-RCequivalent circuit model 42 enables a determination of the OCV 44 duringa real-time drive condition of the vehicle 10. In the 1-RC equivalentcircuit model 42, the resistances 46 and 48 and the capacitance 50 maygenerally be time invariant parameters of the energy storage component14. Alternatively, the OCV 44, which may be used to determine a state ofcharge of the energy storage component 14, may generally be a timevariant parameter of the energy storage component 14. That is, as theenergy storage component 14 is charged and discharged over a time, theOCV 44 will increase and decrease over the time.

An accurate estimation of the OCV 44, the resistances 46 and 48, and thecapacitance 50 may be beneficial to control the energy storage component14 for a longer battery life and increased fuel efficiency ofhybrid-electric vehicles. For example, FIG. 5 illustrates a chart 54that provides a relationship between the OCV 44 and a state of charge(SOC) of the energy storage component 14. The SOC is displayed as apercentage along an abscissa 56 of the chart 54. Additionally, the OCV44 is displayed as a voltage along an ordinate 58 of the chart 54. Acurve 60 represents the relationship between the OCV 44 and the SOC ofthe energy storage component 14. For example, the curve 60 may be usedas a look-up table to provide an accurate SOC representation of theenergy storage component 14. When the OCV 44 of the 1-RC equivalentcircuit model 42 is determined based on measured battery parameters, thevalue of the OCV 44 may be matched with a corresponding SOC percentage.The SOC percentage may provide an operator of the vehicle 10 with anaccurate indication of remaining battery life of the energy storagecomponent 14 in real-time during operation of the vehicle 10. Further,it may be appreciated that the OCV 44 changes as the SOC changes. Forexample, the OCV 44 does not plateau at a voltage when the SOC of theenergy storage component 14 is increasing or decreasing.

Returning to a discussion of FIG. 4, the 1-RC equivalent circuit model42 may be derived initially in discrete time to relate estimations ofthe OCV 44, the resistances 46 and 48, and the capacitance 50 (i.e.,battery parameter estimations) to the measured terminal voltage 52 andthe measured current of the energy storage component 14. A Kalman filteris used for this relationship to determine the battery parameterestimations from the measured terminal voltage 52 and the measuredcurrent. Using the Kalman filter, the control module 32 may update thebattery parameter estimations in real-time with limited reliance onpre-defined battery parameters.

A voltage 52 (e.g., V) of the energy storage component 14 may becalculated by the Duhamel superposition theorem for any arbitrarycurrent source I:

$\begin{matrix}{V = {V_{OC} - {IR}_{0} - {\frac{1}{C_{1}}{\int_{\xi = 0}^{\xi = t}{{I(\xi)}\exp\mspace{11mu}\left( {- \frac{t - \xi}{R_{1}C_{1}}} \right)d\;\xi}}}}} & (1)\end{matrix}$where ξ is a dummy variable of integration. The first two terms on theright side of equation 1 (i.e., V_(OC) and IR₀) give rise to an ohmicdescription of the energy storage component 14, as the voltage 52 isrelated to the OCV 44 reduced by the ohmic drop IR₀. Further, the thirdterm on the right side of equation 1 corresponds to a superpositionintegral, through which past currents influence the OCV 44 beyond thefirst-order effect of changing an average SOC characterizing the energystorage component 14. Because of an exponential weighting function, theimpact of older current-potential data points is exponentially less thanthat of recent data points.

Equation 1 may be evaluated at two arbitrary time steps, t_(k−1) andt_(k), to yield:

$\begin{matrix}{V_{k} - V_{OC} - {I_{k}R_{0}} - {\frac{1}{C_{1}}{\int_{\xi = 0}^{\xi = t_{k}}{{I(\xi)}\exp\mspace{11mu}\left( {- \frac{t_{k} - \xi}{R_{1}C_{1}}} \right)d\;\xi}}}} & (2) \\{and} & \; \\{V_{k - 1} = {V_{OC} - {I_{k - 1}R_{0}} - {\frac{1}{C_{1}}{\int_{\xi = 0}^{\xi = t_{k - 1}}{{I(\xi)}\exp\mspace{11mu}\left( {- \frac{t_{k - 1} - \xi}{R_{1}C_{1}}} \right)d\;\xi}}}}} & (3)\end{matrix}$

It may be assumed that the battery current and voltage are measured at afixed time interval, for example:t _(k−1) −t _(k) ≡Δt, ∀ all k  (4)Accordingly, equations 2 and 3 may be combined to yield:

$\begin{matrix}{{V_{k} = {V_{OC} - {I_{k}R_{0}} - {\exp\mspace{11mu}\left( {- \frac{\Delta\; t}{R_{1}C_{1}}} \right)\left( {V_{OC} - {I_{k - 1}R_{0}} - V_{k - 1}} \right)} - {\frac{1}{C_{1}}{\int_{\xi = t_{k - 1}}^{\xi = t_{k}}{{I(\xi)}\exp\mspace{11mu}\left( {- \frac{t_{k} - \xi}{R_{1}C_{1}}} \right)d\;\xi}}}}}\ } & (5)\end{matrix}$

If (ξ) is approximated by a step current of I_(k−1), equation 5 may bereduced to yield:V _(k) =aV _(k−1)+(1−a)V _(OC) −I _(k) R ₀ −I _(k−1)[(1−a)R ₁ −aR₀]  (6)where

$a = {\exp\mspace{11mu}{\left( {- \frac{\Delta\; t}{R_{1}C_{1}}} \right).}}$Additionally, if I(ξ) is approximated by a step current of(I_(k−1)+I_(k))/2, equation 5 may be reduced to yield:

$\begin{matrix}{V_{k} = {{aV}_{k - 1} + {\left( {1 - a} \right)V_{OC}} - {I_{k}\left\lbrack {{\left( {1 - a} \right)\mspace{11mu}\frac{R_{1}}{2}} + R_{0}} \right\rbrack} - {I_{k - 1}\left\lbrack {{\left( {1 - a} \right)\mspace{11mu}\frac{R_{1}}{2}} - {aR}_{0}} \right\rbrack}}} & (7)\end{matrix}$Further, if I(ξ) is approximated by a piece-wise linear equation of(I_(k−1)+(I_(k)−I_(k−1)))/((t_(k)−t_(k−1))×(ξ−t_(k−1))), equation 5 maybe reduced to yield:

$\begin{matrix}{V_{k} = {{aV}_{k - 1} + {\left( {1 - a} \right)V_{OC}} - {I_{k}\left\lbrack {{\left( {1 + \frac{1 - a}{\ln\mspace{11mu} a}} \right)\mspace{11mu} R_{1}} + R_{0}} \right\rbrack} - {I_{k - 1}\left\lbrack {{{- \left( {a + \frac{1 - a}{\ln\mspace{11mu} a}} \right)}\mspace{11mu} R_{1}} - {a\; R_{0}}} \right\rbrack}}} & (8)\end{matrix}$

Due to the continuous nature of I(ξ) for all time intervals, equation 8may calculate the battery voltage with greater accuracy than equations 6and 7, which are derived based on discontinuous step currents. Forconvenience, equation 8 may be rewritten as:V _(k)=θ₁ −I _(k)θ₃ −I _(k−1)θ₂+θ₄ V _(k−1)  (9)whereθ₁=(1−θ₄)V _(OC)  (10);

$\begin{matrix}{{\theta_{2} = {{{- \left( {\theta_{4} + \frac{1 - \theta_{4}}{\ln\mspace{11mu}\theta_{4}}} \right)}R_{1}} - {\theta_{4}R_{0}}}};} & (11) \\{{\theta_{3} = {{\left( {1 + \frac{1 - \theta_{4}}{\ln\mspace{11mu}\theta_{4}}} \right)R_{1}} + R_{0}}};} & (12) \\{and} & \; \\{\theta_{4} = {\exp\;{\left( {- \frac{\Delta\; t}{R_{1}C_{1}}} \right).}}} & (13)\end{matrix}$

In the present embodiment, equation 9 is used as a battery model torecursively estimate four parameters, θ₁-θ₄, simultaneously from themeasured voltage and current data using a Kalman filter method. Thephysical parameters such as V_(OC) (e.g., OCV 44), R₀ (e.g., resistance46), R₁ (e.g., resistance 48), C₁ (e.g., capacitance 50), and a timeconstant τ=R₁C₁ are extracted from θ₁-θ₄ according to the followingequations:

$\begin{matrix}{{V_{OC} = \frac{\theta_{1}}{1 - \;\theta_{4}}};} & (14) \\{{R_{1} = {- \frac{{\theta_{3}\theta_{4}} + \theta_{2}}{\frac{\left( {1 - \theta_{4}} \right)^{2}}{\ln\mspace{11mu}\theta_{4}}}}};} & (15) \\{{R_{0} = {\theta_{3} + {\frac{{\theta_{3}\theta_{4}} + \theta_{2}}{\frac{\left( {1 - \theta_{4}} \right)^{2}}{\ln\mspace{11mu}\theta_{4}}}\;\left( {1 + \frac{1 - \theta_{4}}{\ln\mspace{11mu}\theta_{4}}} \right)}}};} & (16) \\{and} & \; \\{\tau = {{R_{1}C_{1}} = {- \frac{\Delta\; t}{\ln\mspace{11mu}\theta_{4}}}}} & (17)\end{matrix}$

A standard Kalman filter method may be implemented using a two-modelprocess. To estimate the four parameters, θ₁-θ₄, using the standardKalman filter method, the state transition model may be described by:Θ(k+1)=Θ(k)+R(k)  (18)where R(k) is a process noise vector and Θ(k) is a state vector withθ₁-θ₄ being the four parameters of the state vector. In contrast toequation 18 and other Kalman filter methods, an alternate Kalman filtermethod described below does not calculate state transitions explicitly.Instead, the state measurement mode may be described using the followingequation:V(k)=Φ(k)′Θ(k)+W(k)  (19)where W(k) is a measurement noise and Φ(k) is a regression vectordescribed the following equation:Φ(k)=[1−I(k−1)−I(k)V(k−1)]′  (20)

Further, using the alternate Kalman filter method, a Kalman gain K(k)for SOC estimation is calculated using the following equation:

$\begin{matrix}{{K(k)} = \frac{{P\left( {k - 1} \right)}{\Phi(k)}}{1 + {{\Phi(k)}^{\prime}{P\left( {k - 1} \right)}{\Phi(k)}}}} & (21)\end{matrix}$Accordingly, the state transition model may be updated as follows:Θ(k)=Θ(k−1)+K(k)[y−Φ′Θ(k−1)]  (22)where y is the measured battery voltage 52 at time step k. Further, acovariance matrix P(k) is updated by the following equation:P(k)=[I−K(k)Φ(k)′]P(k−1)+R  (23)

In the alternate Kalman filter method, θ₁ is identified as a fasttime-varying parameter, and θ₂-θ₄ are identified as slow time-varyingparameters. To implement this feature with the alternate Kalman filtermethod, a process noise matrix R is defined with the following equation:

$\begin{matrix}{R = \begin{bmatrix}r & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}} & (24)\end{matrix}$where r is a small value (e.g., r may be set to approximately 0.001)used to control the variability of θ₁ with time using a random walkconcept. This enables the Kalman filter to estimate the time-varyingparameters of a rechargeable battery without using a state transitionmodel, which is often related to calculating the SOC change by currentintegration or coulomb counting. Additionally, by establishing the threeother values across the diagonal of the process noise matrix R(k) atzero, the invariant or slow time-varying parameters, θ₂-θ₄, remaingenerally unchanged.

The computation of the covariance matrix P(k) plays a role in improvingnumerical efficiency and accuracy of the estimated battery parameters.Due to truncation and rounding off errors in a computer, the alternateKalman filter method may lead to a loss of symmetry and positivedefiniteness of P(k), or possibly a divergence. Accordingly, in thealternate Kalman filter method, a U-D factorization method isimplemented for calculating P(k). The U-D factorization method improvesthe positive-definiteness and symmetry of P(k), which may result inattainment of high estimation accuracy and robustness.

Implementing the Kalman filter function with the U-D factorizationmethod may include calculating the following equation:F=U′Φ, G=DF, α ₀=1  (25)Upon calculating equation 25, for the range of values from j=1 to j=M,the following equation is calculated:

$\begin{matrix}\left\{ \begin{matrix}{\alpha = {\alpha_{0} + {{F(j)}{G(j)}}}} \\{{D\left( {j,j} \right)} = {{D\left( {j,j} \right)}{\alpha_{0}/\alpha}}} \\{{b(j)} = {G(j)}} \\{{c(j)} = {{- {F(j)}}/\alpha_{0}}} \\{\alpha_{0} = \alpha}\end{matrix} \right. & (26)\end{matrix}$Additionally, for the range of values from j=1 to j=i−1, the followingequation is calculated:

$\begin{matrix}\left\{ \begin{matrix}{{temp} = {U\left( {i,j} \right)}} \\{{U\left( {i,j} \right)} = {{temp} + {{b(i)} \times {c(j)}}}} \\{{b(i)} = {{b(i)} + {{temp} \times {b(j)}}}}\end{matrix} \right. & (27)\end{matrix}$Further, the value of r is added to D(1,1) with the following equation:D(1,1)=D(1,1)+r  (28)The gain is also calculated with the following equation:K=b/α  (29)Upon completing the calculations for equations 25-29, new estimates arecomputed using the following equation:Θ=Θ+K(y−Φ′Θ)  (30)

Turning now to FIGS. 6A and 6B, a flowchart of a method 70 illustrates amethod for determining battery parameters of the energy storagecomponent 14 using the alternate Kalman filter method described above.Initially, at block 72, initialization is performed by the control unit32. During initialization, arbitrary initial values of the parametersθ₁-θ₄ may be assigned to the state vector Θ. The initial values of theparameters may be arbitrary because the alternate Kalman filter methoduses a starting point to determine the final values of the parameters,and the alternate Kalman filter method does not require a starting pointthat is close to actual values of the parameters θ₁-θ₄. Accordingly, thestate vector Θ may be initialized to [0 0 0 0]′ during initialization.Additionally, during initialization, U, may be initialized as a unitdiagonal matrix (i.e., a matrix with values of 1 assigned to thediagonal elements), and D may be initialized as a diagonal matrix with alarge value (e.g., 1000) assigned to each of the diagonal elements. Thisinitialization may enable the method 70 to begin training a measurementmodel, represented by equation 9 above, when new voltage and currentmeasurements of the energy storage component 14 from the sensors 38become available to the control module 32. Further, the initializationmay include setting values for relative error tolerance (RTOL) and ascaling factor. RTOL may represent a smoothness and flatness criteria,and the scaling factor may be a factor for the battery current to avoidan overflow math error that may be encountered on a fixed-pointmicroprocessor.

Subsequently, as block 74, a data counter may be updated when new databecomes available at the control module 32 from the sensors 38. Forexample, the following equation may represent the data counter:k=k+1  (31)where k is the current data count. Further, at block 76, measured datavalues are assigned to the regression vector Φ(k) at the current andprevious steps. The measured data values may include measured batteryvoltage and current values.

At block 78, the Kalman filter function is executed. The Kalman filterfunction may include the equations 25-30 described above. From theexecution of the Kalman filter function, values of the batteryparameters (e.g., OCV 44, resistances 46 and 48, capacitance 50, andtime constant τ) may be extracted from values of θ₁-θ₄ at block 80.Additionally, at block 82, using the value of the OCV 44 extracted fromvalues of θ₁-θ₄, the SOC of the energy storage component 14 may bedetermined based on an OCV to SOC look-up table stored in a memory 36 ofthe control module 32. The OCV to SOC look-up table may generally bebased on a curve similar to the curve 60 depicted in FIG. 5.

Blocks 84-96 relate to monitoring a convergence of the estimated valuesof the resistances 46 and 48. By monitoring the convergence of theresistances 46 and 48, the control module 32 may confirm that theestimated values of the resistances 46 and 48 are time invariant.Accordingly, at block 84, a mean and a variance of a moving samplewindow of a sample size L of the resistances 46 and 48 are recursivelydetermined. Specifically, the following equations may be used todetermine the means and the variances of the resistances 46 and 48:

if k<=L, then

$\begin{matrix}\left\{ {\begin{matrix}{\mu_{k} = {{\frac{k - 1}{k}\mu_{k - 1}} + {\frac{1}{k}{R_{j}(k)}}}} \\{\sigma_{k}^{2} = {{\frac{k - 2}{k - 1}\sigma_{k - 1}^{2}} + {\frac{k}{\left( {k - 1} \right)^{2}}\left\lbrack {{R_{j}(k)} - \mu_{k}} \right\rbrack}^{2}}} \\{{{X_{j}(k)} = {R_{j}(k)}},{j = 0},1}\end{matrix},} \right. & (32)\end{matrix}$

else

$\begin{matrix}\left\{ \begin{matrix}{i = {{mod}\mspace{11mu}\left( {k,L} \right)}} \\{\mu_{k} = {\mu_{k - 1} + {\frac{1}{N}\left\lbrack {{R_{j}(k)} - {X_{j}(i)}} \right\rbrack}}} \\\begin{matrix}{\sigma_{k}^{2} = {\sigma_{k}^{2} + \frac{{R_{j}(k)}^{2} - {X_{j}(i)}^{2}}{N - 1} -}} \\{\frac{{R_{j}(k)} - {X_{j}(i)}}{N - 1}\left( {{2\mu_{k}} - \frac{{R_{j}(k)} - {X_{j}(i)}}{N}} \right)}\end{matrix} \\{{{X_{j}(k)} = {R_{j}(k)}},{j = 0},1}\end{matrix} \right. & (33)\end{matrix}$where μ_(k) and σ_(k) ² are the sample mean and variance of resistanceR_(j) evaluated at time step k. As seen in equations 32 and 33, therecursive formula developed to calculate the sample mean and variancedoes not involve data copying at each time, and only the new data andthe oldest data are involved in the mean and variance calculations.Thus, the control module 32 calculates equations 32 and 33 efficiently.

Subsequently, at block 86, the control module 32 calculates ratios ofthe variance and a squared mean for each of the resistances 46 and 48,and the control module 32 determines which of the two resistances 46 and48 has a greater ratio. Further, upon determining which ratio isgreater, the control module 32 compares the greater ratio to a square ofthe relative error tolerance (RTOL). This comparison functions as asmoothness and flatness test. The smoothness and flatness test may beused to determine whether the estimated parameters no longer change withtime after a learning time period of the system. Accordingly, to judge aconvergence of the resistances 46 and 48, the smoothness and flatnesstest is passed N times consecutively. The value of N may be 5 cycles, 10cycles, 15 cycles, or any other number of cycles that would reliablyindicate convergence of the resistances 46 and 48. Further, each cycleof the smoothness and flatness test of block 86 may run for a total timeperiod of 0.1 nanoseconds if a sample frequency of 1 second is used forthe method 70.

Subsequently, at block 88, a cycle counter (e.g., iCheck) is updated byadding 1 to the previous cycle counter value if the greater ratio isless than the square of the RTOL. Conversely, if the greater ratio isgreater than the square of the RTOL, at block 90, the cycle counter isreset to zero, and the method 70 returns to initialization at block 72.If the cycle counter is updated at block 88 (i.e., the greater ratio isless than the square of the RTOL), a determination, at block 92, is madeto determine whether the cycle counter has exceeded the value of N. Ifthe cycle counter has not exceeded the value of N, then the method 70returns to initialization at block 72.

However, if the cycle counter has exceeded the value of N, then meanvalues of the converged resistances 46 and 48 and the temperature valueare stored to the memory 36 of the control module 32 at block 94.Through measuring values of the resistances 46 and 48 as functions ofthe temperature, how the resistances 46 and 48 degrade over time may beobserved by the control module 32. Measuring degradation of theresistances 46 and 48 may enable characterization of a state of health(SOH) of the energy storage component 14 from a perspective of ohmicresistance growth of the energy storage component 14. Once the meanvalues of the converged resistances 46 and 48 and the temperature valueare stored, the cycle counter is reset to zero, at block 96, and themethod 70 returns to the initialization at block 72. The method 70 maybe repeated until the control module 32 provides an indication to stopthe operation.

Turning now to FIGS. 7A and 7B, a recursive calculation of sample meansand variances of the resistances 46 and 48 of a sample size L (e.g., asimplemented at block 84 of FIG. 6A) may be stored in a ring buffer 100with L storage positions 102. For example, for a sample size L of 10,the ring buffer 100 may include 10 of the storage positions 102, asdepicted. As new data 104, which includes the sample mean and varianceof the resistances 46 and 48, is calculated, the new data 104 may bestored in one of the storage positions 102. As depicted in FIG. 7A,empty storage positions 102 are populated with the new data 104 inchronological order of when each sample was recorded. Alternatively, asdepicted in FIG. 7B, the new data 104 is stored in the storage position102 of the oldest data value in the ring buffer 100 (e.g., R_(j)(k−9))in such a manner that the new data 104 is stored within the ring buffer100 and old data 106 (e.g., R_(j)(k−9)) is removed from the ring buffer100. Accordingly, only L storage positions 102 are available to storesample means and variances of a sample size L, and the old data 106 isremoved from the ring buffer 100.

There may also be a benefit in estimating a capacity of the energystorage component 14 in real-time. The capacity of the energy storagecomponent 14 may be referred to as a state of health (SOH) of the energystorage component 14. The SOH of the energy storage component 14 may beindicative of a change in a rated capacity of the energy storagecomponent 14. Discussed in detail below are two complementary methodsfor estimating the capacity of the energy storage component 14. A firstmethod, discussed in relation to FIG. 8, provides a linear regression ofreal-time battery current and voltage using a Kalman filter and anequivalent circuit battery model. A second method, discussed in relationto FIG. 9, involves monitoring two open circuit voltage relaxationevents, and calculating the SOH from the two open circuit voltagerelaxation events. Each of the two methods may be executed by thecontrol module 32 to estimate the capacity of the energy storagecomponent.

Turning now to FIGS. 8A and 8B, a method 120 for calculating the SOH ofthe energy storage component 14 is depicted. The method 120 is based ontwo linear regression models running in series at each time step.Because the method 120 relies on linear regression models, the method120 may provide numerical stability for estimating the SOH, asparameters of the linear regression models will converge even wheninitial values of the parameters are arbitrarily selected. Arbitraryselection of the initial values of the parameters may provide a distinctadvantage over extended Kalman filter (EKF) models, which generally relyon accurate initial guesses of parameters of rechargeable batteries dueto nonlinearity of the EKF models. Additionally, the method 120 mayprovide a greater level of tolerance of measurement noises and dataimperfection than the EKF models.

A partial discharge of the energy storage component 14 may be used todetermine the SOH (i.e., the capacity) of the energy storage component14 using the following equation:

$\begin{matrix}{Q = \frac{{100 \times {\int_{0}^{t}{Idt}}}\ }{{SOC} - {SOC}_{0}}} & (34)\end{matrix}$where Q is a capacity of the energy storage component 14 in ampere-hours(Ah), I is the current in amperes, SOC is a current SOC value, and SOC₀is an initial SOC value. As discussed above, the SOC may be obtainedfrom a look-up table similar to the curve 60 of FIG. 5 when an estimatedor known value of the open circuit voltage is available. Equation 34 mayalso be rewritten as the following equation:SOC=SOC₀ +w∫ ₀ ^(t) Idt  (35)where w is equal to 100/Q and ∫₀ ^(t) Idt is an accumulative Ahthroughput of the energy storage component 14. Equation 35 may be usedas a governing equation for battery capacity estimation in the method120. SOC₀ and w are each time-invariant parameters, which can beestimated if the SOC values are known. As discussed above in thediscussion of FIGS. 6A and 6B, the SOC value may be estimated using aKalman filter technique. Accordingly, in the method 120, two linearregression models are used. That is, blocks 122 through 132 maydetermine an SOC value for the energy storage component 14, and blocks134-156 may provide an estimation of SOC₀ and w using the SOC and Ahthroughputs as inputs along with a convergence determination.Accordingly, the battery capacity may be readily calculated from theestimated value of w.

At block 122, the method 120 is initialized. During initialization,arbitrary initial values can be assigned to θ₁-θ₄ for the state vectorΘ. The initial values of the parameters may be arbitrary because thealternate Kalman filter method uses a starting point to determine thefinal values of the parameters, but the alternate Kalman filter methoddoes not rely on a starting point that is close to actual values of theparameters θ₁-θ₄. Accordingly, the state vector Θ may be initialized to[0 0 0 0]′ during an initialization event. Additionally, duringinitialization, U, may be initialized as a unit diagonal matrix (i.e., amatrix with values of 1 assigned to the diagonal elements), and D may beinitialized as a diagonal matrix with a large value (e.g., 1000)assigned to each of the diagonal elements. This initialization mayenable the method 70 to begin training a measurement model, representedby equation 9 above, when new voltage and current measurements of theenergy storage component 14 from the sensors 38 become available to thecontrol module 32. Further, the initialization may include settingvalues for relative error tolerance (RTOL) and a scaling factor. RTOLmay represent a smoothness and flatness criteria, and the scaling factormay be a factor for the battery current to avoid an overflow math errorthat may be encountered on a fixed-point microprocessor. Initializationat block 122 may be performed on both the linear regression model forthe SOC estimation and the linear regression model for the SOC₀ and w.

At block 124, a data counter may be updated when new data becomesavailable at the control module 32 from the sensors 38. For example,equation 31 may be used as a representation of the current data count.Additionally, an ampere-hour throughput may also be updated when databecomes available using the following equation:q=q+Δq _(k)  (36)where q is the ampere-hour throughput. Further, at block 126, measureddata values are assigned to the regression vector Φ(k) at the currentand previous steps. The measured data values may include measuredbattery voltage and current values.

At block 128, the Kalman filter function is executed. The Kalman filterfunction may include the equations 25-30 described above. From theexecution of the Kalman filter function, values of the OCV 44, inaddition to the resistances 46 and 48, capacitance 50, and time constantτ, may be extracted from values of θ₁-θ₄ at block 130. Additionally, atblock 132, using the value of the OCV 44 extracted from values of θ₁-θ₄,the SOC of the energy storage component 14 may be determined based on anOCV to SOC look-up table stored in a memory 36 of the control module 32.The OCV to SOC look-up table may generally be based on a curve similarto the curve 60 depicted in FIG. 5.

Subsequently, at block 134, the control module 32 makes a determinationof whether the estimated SOC is usable as an input to proceed with thesubsequent capacity estimation of the energy storage component 14. Forexample, the control module 32 may make the determination based onwhether a battery terminal temperature requirement has been met (e.g.,the temperature of a terminal of the energy storage component is lessthan approximately 25 degrees Celsius) and whether a minimum SOClearning time period has been exceeded (e.g., the time period betweenthe current measurement and the initial condition is greater than 100seconds). If either of these conditions are not met, at block 136, thecurrent and voltage measurements are reset to the previous values, andthe method 120 restarts at the data counter update of block 124.

If the SOC is available for use in the capacity estimation, at block138, minimum and maximum SOC values are tracked for use in calculatingthe maximum SOC swing. Further, the ampere-hour throughput is assignedto a second regression vector at block 140. The second regression vectoris represented by Φ1 and represents the following equation:Φ1(k)=[1q/scale 1]′  (37)where q is the ampere hour throughput and scale 1 is a scaling factorfor the estimated capacity to avoid an overflow math error whencalculated on a fixed-point microprocessor.

Upon assigning the ampere-hour throughput to the second regressionvector, the Kalman filter function is executed at block 142. Executingthe Kalman filter function may update the covariance matrix U1 and D1and the parameter vector Θ1 for the capacity estimation, which wasinitialized as [0 0 0 0]′ along with the parameter vector Θ for the SOCestimation. Further, the Kalman gain K(k) for the capacity estimation isrepresented by the following equation:

$\begin{matrix}{{K\; 1(k)} = \frac{P\; 1\left( {k - 1} \right)\Phi\; 1(k)}{1 + {\Phi\; 1(k)^{\prime}P\; 1\left( {k - 1} \right)\Phi\; 1(k)}}} & (38)\end{matrix}$And the covariance matrix P1(k) is updated by the following equation:P1(k)=[I−K1(k)Φ1(k)′]P1(k−1)  (39).When compared to equation 23 of the SOC estimation, equation 39 does notinclude the process noise matrix R because both SOC₀ and w aretime-invariant parameters.

Subsequently, at block 144, an estimated capacity value Q may beextracted from the parameter vector Θ1. For example, the estimatedcapacity value Q may be represented by the following equation:Q=−100/θ1₂  (40)where θ1₂ is the second parameter from the parameter vector Θ1. Afterextracting the estimated capacity value Q, blocks 146-156 may be used tomonitor convergence of the capacity estimation.

Accordingly, at block 146, a mean and a variance of a moving samplewindow of a sample size L of the estimated capacity value Q isrecursively determined and stored in a ring buffer similar to the ringbuffer 100 discussed above. After the mean and variance are determined,at block 148, the variance to mean squared ratio is compared to thesquare of the RTOL. This comparison functions as a smoothness andflatness test. The smoothness and flatness test may be used to determinewhether the estimated parameters (e.g., the estimated capacity value Q)no longer change with time after a learning time period of the system.Accordingly, to judge a convergence of the estimated capacity value Q,the smoothness and flatness test is passed N times consecutively. Thevalue of N may be 5 cycles, 10 cycles, 15 cycles, or any other number ofcycles that would reliably indicate convergence of the estimatedcapacity value Q. Further, each cycle of the smoothness and flatnesstest of block 148 may run for a total time period of 0.1 nanoseconds ifa sample frequency of 1 second is used for the method 120.

Subsequently, at block 150, a cycle counter (e.g., iCheck) is updated byadding 1 to the previous cycle counter value if the ratio is less thanthe square of the RTOL. Conversely, if the ratio is greater than thesquare of the RTOL, at block 136, the current and voltage measurementsare reset to the previous values, and the method 120 restarts at thedata counter update of block 124. If the cycle counter is updated atblock 150 (i.e., the greater ratio is less than the square of the RTOL),a determination, at block 152, is made to determine whether the cyclecounter has exceeded the value of N and whether the SOC swing (maximumSOC minus the minimum SOC) is greater than 20% of SOC₀. A threshold usedat block 152 for the SOC swing may also be another percentage of SOC₀depending on the specific energy storage component 14. For example, thepercentage could be 25% or up to 50%, or the percentage could be 10% or15% of SOC₀. If the cycle counter has not exceeded the value of N and/orthe SOC swing is not large enough, then the method 120 returns to block136 to reset voltage and current values.

If the cycle counter has exceeded the value of N and the SOC swing issufficient, at block 154, the extracted estimated capacity value Q issaved to the memory 36. After the estimated capacity value Q is saved tothe memory 36, the cycle counter is reset to zero, and the method 120returns to block 136. Additionally, the method 120 may repeat asdescribed above until the control module 32 receives an indication tostop estimating the estimated capacity value Q of the energy storagecomponent 14.

FIGS. 9A and 9B depict a method 160 for calculating the SOH of theenergy storage component 14 by monitoring two valid open circuit voltage(OCV) relaxation events and integrating current over time between thetwo valid OCV relaxation events. A first of the two valid OCV relaxationevents may be measured immediately upon starting the vehicle 10 after anextended rest period (e.g., after the vehicle has been turned off forgreater than 2 hours), and a second of the two valid OCV relaxationevents may be measured when the energy storage component 14 has beenrelaxed for an extended period (e.g., after the vehicle 10 is turned offand a predetermined amount of time has passed). Additionally, a timebetween the two valid OCV relaxation events may be limited to limit anaccumulation of current offset error and improve accuracy of the SOHmeasurement.

At block 162, the method 160 is initialized. During initialization, timecounters t₁ and t₂, starting ampere-hour throughput Σq, and a binary OCVvalidity status iOCV are set. If the OCV of the energy storage component14 is fully relaxed at the moment the control module 32 wakes up (e.g.,when the vehicle 10 is cranked), a value of iOCV may be set to 1 duringinitialization. Otherwise, the value of iOCV may be set to 0 duringinitialization. The OCV may be fully relaxed after parking a vehicle foran extended period of time (e.g., greater than one hour) at roomtemperature with a negligible current drain value of less than arelaxation current threshold I_(Relax). One way to confirm whether theOCV is fully relaxed is to monitor the rate of OCV change over time, asthe OCV may change asymptotically with time. After an extended period oftime at room temperature, the OCV may change very slowly with time, andsuch an OCV may be used to represent the relaxed OCV with a sufficientaccuracy for the method 160.

After initialization, the counter is updated, at block 164, when newdata becomes available at the control module 32 from the sensors 38. Forexample, equation 31 may be used as a representation of the current datacount. Upon updating the counter, at block 166, data is read from thesensors 38 of the energy storage component 14. The data may includebattery current, voltage, and a step change in ampere-hour throughput.Accordingly, with the step change in ampere-hour throughput, at block168, an accumulative ampere-hour throughput is updated using thefollowing equation:Σq=Σq+Δq _(k)  (41)where Σq represents the accumulative ampere-hour throughput, and Δq_(k)represents the step change in ampere-hour throughput.

At block 170, a determination is made by the control module 32 as towhether the battery current meets the predetermined relaxation currentthreshold I_(Relax). The determination may be accomplished by measuringan absolute value of the battery current at the current step, andcomparing the absolute value to I_(Relax). If the absolute value of thebattery current is greater than I_(Relax), then the battery is notsufficiently relaxed. At this juncture, the time step may be monitored,and the iOCV value may remain at 0 or be set to 0 at block 172.Accordingly, the method 160 may start again at the counter update ofblock 164.

Alternatively, if the absolute value of the battery current is less thanI_(Relax), then a determination of the iOCV validity status may bedetermined at block 174. That is, if the value of iOCV is 1, then theenergy storage component 14 has been in a relaxed state for a sufficientamount of time, and the OCV may be set to the current voltage readingV_(k), at block 178. Alternatively, if the iOCV is 0, then, at block176, a determination as to whether a sufficient amount of time haspassed for the energy storage component 14 to be sufficiently relaxed ismade. For example, an amount of time from initialization to the currentstep may be compared to a relaxation threshold time t_(Relax). Asmentioned above, t_(Relax) may be set to 1 hour, 2 hours, or more hoursunder standard operating conditions. However, t_(Relax) may increase ordecrease based on external conditions, such as the battery temperature.In general, the greater the temperature of the battery, the shorter thet_(Relax) time may be. If the relaxation threshold time t_(Relax) hasnot been met, then the method 160 returns to the counter update at block164 to begin the method 160 again.

Alternatively, if the relaxation threshold time t_(Relax) has been met,then the energy storage component 14 has been in a relaxed state for asufficient amount of time, and the OCV may be set to the current voltagereading V_(k), at block 178. From the OCV value determined from thecurrent voltage reading V_(k), the current SOC may be determined from anOCV versus SOC lookup table similar to the curve 60 of FIG. 5.

Subsequently, at block 182, a determination is made as to whether thecounter value k is greater than the sample size L. If the counter valuek is greater than L, a value of a modulus variable M is updated byadding one to the value at block 184. The modulus variable M representsa cumulative number of times the OCV criteria has been met.Alternatively, if the counter value k is not greater than the samplesize L, then the value of L is set to the counter value k plus 1 atblock 186. After updating the values of the modulus variable M and/orthe sample size L, a modulus operation is performed, at block 188, toalternate recording the current SOC value between two memory storagelocations within the memory 36 of the control module 32. Accordingly,the method 160 will alternate storing the current SOC value at SOC₁, atblock 190, and SOC₂ at block 192. Additionally, the time andaccumulative ampere-hour throughput may also be stored in memory atblocks 190 and 192.

At block 194, a determination is made as to whether there is asufficient SOC swing between the value stored for SOC₁ and the valuestored for SOC₂. The sufficient SOC swing may be described as SOC_(MIN),and SOC_(MIN) may represent a threshold percentage difference betweenthe two stored SOC values. For example, SOC_(MIN) may be a 5% swing, a10% swing, a 15% swing, or another percentage swing that establishes asufficient difference between the two SOC values for an accuratecapacity estimation of the energy storage component 14. If the SOC swingis not sufficient, the method 160 may return to the counter update atblock 164.

If the SOC swing is sufficient, a determination may be made by thecontrol module 32 as to whether a total current integration time is lessthan a maximum allowable time, at block 196. For example, if the totalcurrent integration time is greater than the maximum allowable time, acurrent offset error may impact the capacity estimation in anundesirable manner. Accordingly, it may be desirable to limit the totalcurrent integration time to less than approximately 50 hours. If themaximum allowable time has been exceeded, the method 160 may return tothe counter update at block 164.

Alternatively, if the maximum allowable time has not been exceeded, theestimated battery capacity Q of the energy storage component 14 may becalculated at block 198. To calculate the estimated battery capacity Q,the following equation may be used:

$\begin{matrix}{Q = {{- \frac{{\sum\;{q\; 2}} - {\sum\;{q\; 1}}}{{{SOC}\; 2} - {{SOC}\; 1}}} \times 100}} & (42)\end{matrix}$where the values used for the calculation are obtained from the valuesstored at blocks 190 and 192. Subsequently, the estimated batterycapacity Q is stored to the memory 36 at block 200. Further, uponstoring the estimated battery capacity Q, the method 160 may return tothe counter update of block 164, and the method 160 may operaterecursively until an indication is provided to the control module 32 tostop the method 160.

The two capacity estimation methods 120 and 160 described above arecomplimentary to each other in practice. The method 120 may not use SOCswing between two OCV relaxation events, but may use real-time SOC swingwhile the vehicle 10 is in operation. Accordingly, the method 120 may beparticularly applicable to advance start-stop and hybrid electricvehicles due to a design principle of the advance start-stop and hybridelectric vehicles to maximize charge rate to harvest energy when thereis a surplus supply of kinetic energy and to maximize a discharge rateto improve fuel economy when there is a peak power consumption demand.Accordingly, the energy storage component 14 may establish a standby(rest) mode at an SOC of around 50%. Therefore, the energy storagecomponent 14 would generally have a small separation between two OCVmeasurements. Alternatively, due to its simplicity, accuracy, and lowimplementation cost, the method 160 may be applicable when the vehicle10 experiences several rest periods during typical operation to enhancethe robustness and accuracy of the capacity estimations of the energystorage component 14.

It may be beneficial to occasionally verify capacity estimation resultsof the energy storage component 14. Accordingly, FIG. 10 is a flowchart210 describing a verification process of the estimated battery capacityQ of the energy storage component 14. At block 212, a determination ismade by the control module 32 as to whether the capacity estimation forthe energy storage component 14 is valid. For example, the capacityestimation may be invalid if there have not been any valid capacityestimation results from the methods 120 and 160 for an extended amountof time (e.g., greater than two months since the last valid capacityestimation results), if a recently calculated capacity estimationexceeds a threshold of change (e.g., the recently calculated capacityestimation is greater than 5 percent, greater than 10 percent, greaterthan 15 percent, or more different from the previously calculatedcapacity estimation), or if accuracy of a valid capacity estimation isnot within an acceptable range. When any of these criteria are met, thecontrol module 32 may initialize the verification process of theflowchart 210.

In determining if the accuracy of a valid capacity estimation is notwithin an acceptable range, an error of the capacity estimation may becalculated using an error of the SOC/OCV measurements as well as viacurrent integration assuming current measurement accuracy. For example,Kalman filter methods may include an error estimation or maximumaccuracy estimation (e.g., approximately 3%), and evaluation of theconvergence criteria, as described herein, may refine this errorestimation. After refining the error estimation, a time or energythroughput dependent error increase based on the expected aging underobserved conditions of the battery system 12 may be applied to the errorestimation. Accordingly, if a capacity estimation is determined within acertain accuracy, the error steadily increases until an opportunity fora new estimation with a lower error is reached, which will reset theerror estimation.

If the capacity estimation is determined to be valid, then the currentcapacity estimation may be updated at block 214. The capacity updated atblock 214 may be used, at block 216, as an actual capacity of the energystorage component 14 during a current integration process using thecurrent capacity estimation (e.g., using equation 35, above). The resultof the current integration process with the actual capacity of theenergy storage component is a value of SOC(1) over an integration timeperiod that is calculated at block 218.

Alternatively, if the capacity estimation is determined to not be valid(e.g., a mean error is too high, and a reset would not improveaccuracy), then a candidate capacity may used, at block 220, in acurrent integration process that is parallel to the current integrationprocess of block 216. The candidate capacity may be 5 percent less thanthe actual capacity used in the current integration process of block216. The result of the current integration process using the candidatecapacity is a value of SOC(2) over the integration time period that iscalculated at block 222. Additionally, in performing the currentintegration processes at blocks 216 and 220, measured parameters 224 ofthe energy storage component 14 may be used. The measured parameters 224may include system/sensor specifications, temperature, current, andvoltage of the energy storage component 14.

To effectively use the verification process of the flowchart 210, anaccurate initial SOC at the start of a parallel current integrationprocess may be used. Therefore, the measured parameters 224 may providevalues for the battery control module 32 to calculate the accurateinitial SOC. For example, if the vehicle 10 exits an extended restperiod of the energy storage component 14 prior to the parallel currentintegration processes at blocks 216 and 220, the open circuit voltage 44of the energy storage component 14 may be measured from the measuredparameters 224, as discussed in detail above. Using the open circuitvoltage 44, an open circuit voltage to state of charge look-up tablestored in the memory 36 may be consulted by the control module 32 tocalculate an accurate initial SOC value. It may also be appreciated thata significant SOC delta (e.g., an SOC swing) between an initial SOC at astart of the parallel current integration process and a final SOC at apoint in time where a changing capacity estimation value is assessed maybe beneficial for accurate assessment of the validity of the capacityestimation value. This value may be represented by ΔSOC^(δQ).

Additionally, to compare to the SOC values resulting from the parallelcurrent integration processes, a Kalman filter at block 226 may be usedat the point in time where the changing capacity estimation value isassessed. For example, using the method 70 discussed above relating toFIGS. 6A and 6B, an SOC estimation (SOC(3)) of the energy storagecomponent 14 may be calculated by the control module 32 at block 228.Alternatively, or in addition, if the energy storage component 14 is ina rest state, the OCV 44 may be obtained from the energy storagecomponent 14 at block 230. Consultation by the control module 32, atblock 232, of an OCV to SOC look-up table results in a calculation ofthe SOC(4) at block 234.

After SOC(1)-SOC(3) and/or SOC(4) are calculated, a directionalcomparison of the SOC values may be performed by the control module 32at block 236. For example, the values of SOC(3) and SOC(4) at the pointin time where the changing capacity estimation value is assessed may becompared to the values of SOC(1) and SOC(2) at the same point in time.This comparison may provide a number of details about the new capacityestimation of the energy storage component 14. If the values of SOC(3)and/or SOC(4) fall within the values of SOC(1) and SOC(2) or the valuesof SOC(3) and/or SOC(4) are in the same direction from SOC(1) as thevalue of SOC(2), then the control module 32 may deem an estimatedcapacity calculated from SOC(3) and/or SOC(4) to be valid. In such asituation, the control module 32 may calculate the capacity at thispoint in time, and update the actual capacity value at block 214 to thenewly calculated estimated capacity.

Alternatively, if the values of SOC(3) and/or SOC(4) do not fall withinthe values of SOC(1) and SOC(2), then the control module 32 may deem acapacity estimation from SOC(3) and/or SOC(4) invalid at block 236. Insuch a situation, the process of the flowchart 210 may restart at adifferent initial SOC value, or SOC(3) and/or SOC(4) may be calculatedat a later time to determine whether a capacity value based on SOC(3)and/or SOC(4) is valid.

Turning to FIG. 11, a chart 240 illustrates the process detailed in theflowchart 210 of FIG. 10. An ordinate 242 represents the SOC of theenergy storage component 14 as a percentage. An abscissa 244 representsa time against which the SOC is measured. Line 246 represents thecurrent integration of the energy storage component 14 using the actualcapacity of the energy storage component, and line 248 represents theparallel current integration of the energy storage component 14 usingthe candidate capacity of the energy storage component 14, which, asillustrated, is five percent lower than the actual capacity of theenergy storage component 14.

As illustrated, the parallel current integration process begins at timet₁ with an accurate initial SOC value 250. The accurate initial SOCvalue 250 may be obtained with an open circuit voltage (OCV) measurementand a comparison of the OCV measurement with an OCV to SOC look-uptable. Additionally, the parallel current integration process may beperformed until a time t₂. The time t₂ may represent a time at which aΔSOC^(Q) (e.g., a state of charge swing) of the parallel currentintegration process is sufficient for an accurate verification of theestimated capacity of the energy storage component 14. For example, thestate of charge swing may be a swing of approximately 20 percent foraccurate verification of the estimated capacity. In other instances, thestate of charge swing may be a swing of approximately 10 percent, 15percent, or up to 25 percent or greater for accurate verification of theestimated capacity.

As the parallel integration process approaches time t₂, several SOCmeasurements 252 and 254 of the energy storage component calculatedusing a Kalman filter method, for example, are plotted. The dark coloredSOC measurements 252 represent SOC measurements that are not valid foruse with a capacity estimation update. For example, changes in the SOC(e.g., ΔSOC^(k)) between the line 246 and the SOC measurements 252 arein the opposite direction compared to the SOC swing (e.g., ΔSOC^(δQ)) ofthe parallel current integration lines 246 and 248. Accordingly, if thevalues of the dark colored SOC measurements 252 were calculated by thecontrol module 32, the control module 32 would wait for a subsequentvalid SOC measurement to update the capacity estimation.

Alternatively, the light colored SOC measurements 254 represent SOCmeasurements that are valid for use with a capacity estimation update.For example, changes in the SOC (e.g., ΔSOC^(k)) between the line 246and the SOC measurements 254 are in the same direction as the SOC swing(e.g., ΔSOC^(δQ)) of the parallel current integration lines 246 and 248.Accordingly, if the values of the light colored SOC measurements 254were calculated by the control module 32, the control module 32 wouldupdate the capacity estimation at the time that the light colored SOCmeasurements 254 were calculated.

One or more of the disclosed embodiments, alone or in combination, mayprovide one or more technical effects including determining battery timevariant and time invariant variables, determining a state of charge ofthe battery, determining a state of health of the battery, andvalidating the state of health of the battery. The technical effects andtechnical problems in the specification are exemplary and are notlimiting. It should be noted that the embodiments described in thespecification may have other technical effects and can solve othertechnical problems.

While only certain features and embodiments have been illustrated anddescribed, many modifications and changes may occur to those skilled inthe art (e.g., variations in sizes, dimensions, structures, shapes andproportions of the various elements, values of parameters (e.g.,temperatures, pressures, etc.), mounting arrangements, use of materials,colors, orientations, etc.) without materially departing from the novelteachings and advantages of the disclosed subject matter. The order orsequence of any process or method steps may be varied or re-sequencedaccording to alternative embodiments. It is, therefore, to be understoodthat the appended claims are intended to cover all such modificationsand changes as fall within the true spirit of the disclosure.Furthermore, in an effort to provide a concise description of theexemplary embodiments, all features of an actual implementation may nothave been described. It should be appreciated that in the development ofany such actual implementation, as in any engineering or design project,numerous implementation specific decisions may be made. Such adevelopment effort might be complex and time consuming, but wouldnevertheless be a routine undertaking of design, fabrication, andmanufacture for those of ordinary skill having the benefit of thisdisclosure, without undue experimentation.

The invention claimed is:
 1. An automotive vehicle comprising: abattery, wherein the battery comprises a plurality of battery cells; anelectrical system comprising an electrical generator configured to, inoperation, generate electrical power and an electrical load configuredto operate using electrical power; a relay electrically coupled betweenthe plurality of battery cells and the electrical system; one or moresensors configured to measure a plurality of battery parameter setsduring operation of the battery, wherein each battery parameter set ofthe plurality of battery parameter sets comprises a terminal voltage anda terminal current; and one or more control modules communicativelycoupled to the relay and the one or more sensors, wherein the one ormore control modules are configured to: determine a first Kalman filterbased at least in part on a first relationship between the terminalvoltage and the terminal current of the battery with an open circuitvoltage of the battery, an ohmic resistance of the battery, and a chargetransfer resistance of the battery; determine a second Kalman filterbased at least in part on a second relationship between capacity of thebattery, a change in state of charge of the battery, and an accumulativeampere-hour throughput of the battery during the change in the state ofcharge; determine a plurality of estimated open circuit voltages byexecuting the first Kalman filter based at least in part on each batteryparameter set of the plurality of battery parameter sets; determine aplurality of estimated states of charge each corresponding with one ofthe plurality of estimated open circuit voltages based at least in parton a third relationship between the state of charge and the open circuitvoltage of the battery; determine the capacity of the battery based atleast in part on execution of the second Kalman filter using eachestimated state of charge of the plurality of estimated states ofcharge; and control charging of the battery, discharging of the battery,or both based at least in part on the capacity of the battery by:instructing the relay to switch to, maintain, or both a connected stateto enable the electrical generator to charge the battery, the battery todischarge electrical power to the electrical load, or both; andinstructing the relay to switch to, maintain, or both a disconnectedstate to block the electrical generator from charging the battery andthe battery from discharging to the electrical load.
 2. The automotivevehicle of claim 1, wherein the plurality of battery cells comprises: afirst battery cell that uses a first battery chemistry; and a secondbattery cell that uses a second battery chemistry.
 3. The automotivevehicle of claim 1, wherein the one or more control modules areconfigured to: determine a plurality of estimated capacities byexecuting the second Kalman filter using each estimated state of chargeof the plurality of estimated states of charge; determine a meanestimated capacity of the plurality of estimated capacities; determinean estimated capacity variance of the plurality of estimated capacitiesof the battery; determine whether a ratio of the estimated capacityvariance to the mean estimated capacity squared is greater than arelative error tolerance threshold; and determine the capacity of thebattery based at least in part on the plurality of estimated capacitieswhen the ratio of the estimated capacity variance to the mean estimatedcapacity squared is not greater than the relative error tolerancethreshold.
 4. The automotive vehicle of claim 3, wherein the one or morecontrol modules are configured to: determine a maximum estimated stateof charge from the plurality of estimated states of charge; determine aminimum estimated state of charge from the plurality of estimated stateof charge; determine whether a difference between the maximum estimatedstate of charge and the minimum estimated state of charge is greaterthan a swing threshold; and determine the capacity of the battery basedat least in part on the plurality of estimated capacities when the ratioof the estimated capacity variance to the mean estimated capacitysquared is not greater than the relative error tolerance threshold andthe difference between the maximum estimated state of charge and theminimum estimated state of charge is greater than the swing threshold.5. The automotive vehicle of claim 4, wherein the one or more controlmodules are configured to: determine a most recent open circuit voltageof the plurality of estimated open circuit voltages by executing thefirst Kalman filter based at least in part on a most recent batteryparameter set of the plurality of battery parameter sets; determine amost recent estimated state of charge of the plurality of estimatedstates of charge by mapping the most recent open circuit voltage to themost recent estimated state of charge based at least in part on thethird relationship between the state of charge and the open circuitvoltage of the battery; determine a most recent estimated capacity ofthe plurality of estimated capacities by executing the second Kalmanfilter based at least in part on the most recent estimated state ofcharge; and set the capacity of the battery equal to the most recentestimate capacity when the ratio of the estimated capacity variance tothe mean estimated capacity squared is not greater than the relativeerror tolerance threshold and the difference between the maximumestimated state of charge and the minimum estimated state of charge isgreater than the swing threshold.
 6. The automotive vehicle of claim 1,wherein the one or more control modules are configured to: determine theaccumulative ampere-hour throughput of the battery based at least inpart on the terminal current indicated in each battery parameter set ofthe plurality of battery parameter sets; and determine the capacity ofthe battery based at least in part on execution of the second Kalmanfilter using the accumulative ampere-hour throughput of the battery. 7.The automotive vehicle of claim 1, wherein the one or more controlmodules are configured to: initialize parameters of the first Kalmanfilter with random values before execution of the first Kalman filterbased at least in part the plurality of battery parameter sets; andinitialize parameters of the second Kalman filter with random valuesbefore execution of the second Kalman filter based at least in part onthe plurality of estimated states of charge.
 8. The automotive vehicleof claim 1, wherein: the one or more control modules are communicativelycoupled to the electrical generator; and the one or more control modulesare configured to control charging of the battery, discharging of thebattery, or both by controlling electrical power output by theelectrical generator.
 9. The automotive vehicle of claim 8, wherein theone or more control modules comprise: a battery control unit configuredto control switching of the relay; and a vehicle control unit configuredto control electrical power output by the electrical generator.
 10. Amethod comprising: receiving, using one or more processors implementedin a vehicle, a plurality of battery parameter sets measured by one ormore sensors during operation of a battery implemented in the vehicle,wherein each battery parameter set of the plurality of battery parametersets comprises a terminal voltage and a terminal current; determining,using the one or more processors, a first Kalman filter based at leastin part on a first relationship between the terminal voltage and theterminal current of the battery with an open circuit voltage of thebattery, an ohmic resistance of the battery, and a charge transferresistance of the battery; determining, using the one or moreprocessors, a second Kalman filter based at least in part on a secondrelationship between capacity of the battery, a change in state ofcharge of the battery, and an accumulative ampere-hour throughput of thebattery during the change in the state of charge; determining, using theone or more processors, a plurality of estimated open circuit voltagesby executing the first Kalman filter based at least in part on eachbattery parameter set of the plurality of battery parameter sets;determining, using the one or more processors, a plurality of estimatedstates of charge each corresponding with one of the plurality ofestimated open circuit voltages based at least in part on a thirdrelationship between the state of charge and the open circuit voltage ofthe battery; determining, using the one or more processors, the capacityof the battery based at least in part on execution of the second Kalmanfilter using each estimated state of charge of the plurality ofestimated states of charge; and controlling, using the one or moreprocessors, charging of the battery, discharging of the battery, or bothbased at least in part on the capacity of the battery, whereincontrolling charging of the battery, discharging of the battery, or bothcomprises: instructing a relay electrically coupled between the batteryand an electrical system implemented in the vehicle to switch to,maintain, or both a connected state to enable the electrical system tocharge the battery, the battery to discharge electrical power to theelectrical system, or both; and instructing the relay to switch to,maintain, or both a disconnected state to block the electrical systemfrom charging the battery and the battery from discharging to theelectrical system.
 11. The method of claim 10, wherein determining thecapacity of the battery comprises: determining a plurality of estimatedcapacities by executing the second Kalman filter using each estimatedstate of charge of the plurality of estimated states of charge;determining a mean estimated capacity of the plurality of estimatedcapacities; determining an estimated capacity variance of the pluralityof estimated capacities of the battery; determining whether a ratio ofthe estimated capacity variance to the mean estimated capacity squaredis greater than a relative error tolerance threshold; and determiningthe capacity of the battery based at least in part on the plurality ofestimated capacities when the ratio of the estimated capacity varianceto the mean estimated capacity squared is not greater than the relativeerror tolerance threshold.
 12. The method of claim 11, whereindetermining the capacity of the battery comprises: determining a maximumestimated state of charge from the plurality of estimated states ofcharge; determining a minimum estimated state of charge from theplurality of estimated state of charge; and determining whether adifference between the maximum estimated state of charge and the minimumestimated state of charge is greater than a swing threshold; anddetermining the capacity of the battery based at least in part on theplurality of estimated capacities when the ratio of the estimatedcapacity variance to the mean estimated capacity squared is not greaterthan the relative error tolerance threshold and the difference betweenthe maximum estimated state of charge and the minimum estimated state ofcharge is greater than the swing threshold.
 13. The method of claim 12,wherein determining the capacity of the battery comprises: determining amost recent open circuit voltage of the plurality of estimated opencircuit voltages by executing the first Kalman filter based at least inpart on a most recent battery parameter set of the plurality of batteryparameter sets; determining a most recent estimated state of charge ofthe plurality of estimated states of charge by mapping the most recentopen circuit voltage to the most recent estimated state of charge basedat least in part on the third relationship between the state of chargeand the open circuit voltage of the battery; determining a most recentestimated capacity of the plurality of estimated capacities by executingthe second Kalman filter based at least in part on the most recentestimated state of charge; and setting the capacity of the battery equalto the most recent estimate capacity when the ratio of the estimatedcapacity variance to the mean estimated capacity squared is not greaterthan the relative error tolerance threshold and the difference betweenthe maximum estimated state of charge and the minimum estimated state ofcharge is greater than the swing threshold.
 14. The method of claim 10,comprising: initializing, using the one or more processors, parametersof the first Kalman filter with random values before execution of thefirst Kalman filter based at least in part the plurality of batteryparameter sets; and initializing, using the one or more processor,parameters of the second Kalman filter with random values beforeexecution of the second Kalman filter based at east least in part on theplurality of estimated states of charge.
 15. The method of claim 10,wherein controlling charging of the battery, discharging of the battery,or both comprises controlling voltage, current, or both of electricalpower output by an electrical generator implemented in the electricalsystem.
 16. A tangible, non-transitory, computer-readable medium storinginstruction executable by one or more processors, wherein theinstructions comprise instructions to: receive, using the one or moreprocessors, a plurality of battery parameter sets measured by one ormore sensors during operation of a battery, wherein the plurality ofbattery parameter sets comprises: a first battery parameter set measuredat a first time, wherein the first battery parameter set comprises afirst terminal voltage of the battery and a first terminal current ofthe battery; and a second battery parameter set measured at a secondtime after the first time, wherein the second battery parameter setcomprises a second terminal voltage of the battery and a second terminalcurrent of the battery; determine, using the one or more processors,whether the battery is relaxed at the first time based at least in parton the first battery parameter set and whether the battery is relaxed atthe second time based at least in part on the second battery parameterset; when the battery is relaxed at the first time and at the secondtime: determine, using the one or more processors, a first state ofcharge of the battery present at the first time by mapping the firstterminal voltage based on a first relationship between state of chargeand open circuit voltage of the battery; determine, using the one ormore processors, a second state of charge of the battery present at thesecond time by mapping the second terminal voltage based on the firstrelationship between state of charge and open circuit voltage of thebattery; determine, using the one or more processors, an accumulativeampere-hour throughput of the battery between the first time and thesecond time based at least in part on one or more of the plurality ofbattery parameter sets measured between the first time and the secondtime; and determine, using the one or more processors, capacity of thebattery based at least in part on the accumulative ampere-hourthroughput of the battery between the first time and the second time anda difference between the first state of charge of the battery and thesecond state of charge of the battery; and control, using the one ormore processors, charging of the battery, discharging of the battery, orboth based at least in part on the capacity of the battery by:instructing a relay electrically coupled between the battery and anelectrical system to switch to, maintain, or both a connected state toenable the electrical system to charge the battery, the battery todischarge electrical power to the electrical system, or both; andinstructing the relay to switch to, maintain, or both a disconnectedstate to block the electrical system from charging the battery and thebattery from discharging to the electrical system.
 17. The tangible,non-transitory, computer-readable medium of claim 16, wherein: theinstructions to determine whether the battery is relaxed at the firsttime comprise instructions to determine that the battery is relaxed atthe first time when the first terminal current measured at the firsttime does not exceed a relaxation current threshold; and theinstructions to determine whether the battery is relaxed at the secondtime comprise instructions to determine that the battery is relaxed atthe second time when the second terminal current measured at the secondtime does not exceed the relaxation current threshold.
 18. The tangible,non-transitory, computer-readable medium of claim 16, wherein: theinstructions to determine whether the battery is relaxed at the firsttime comprise instructions to: determine a first rate of change of theopen circuit voltage of the battery based at least in part on the firstterminal voltage measured at the first time; and determine that thebattery is relaxed at the first time when the first rate of change ofthe open circuit voltage is less than a rate of change threshold; andthe instructions to determine whether the battery is relaxed at thesecond time comprise instructions to: determine a second rate of changeof the open circuit voltage of the battery based at least in part on thesecond terminal voltage measured at the second time; and determine thatthe battery is relaxed at the second time when the second rate of changeof the open circuit voltage is less than the rate of change threshold.19. The tangible, non-transitory, computer-readable medium of claim 16,comprising instructions to, when the battery is not relaxed at the firsttime and not relaxed at the second time: determine, using the one ormore processors, a first Kalman filter based at least in part on asecond relationship between a terminal voltage of the battery, aterminal current of the battery, the open circuit voltage of thebattery, an ohmic resistance of the battery, and a charge transferresistance of the battery; determine, using the one or more processors,a second Kalman filter based at least in part on a third relationshipbetween the capacity of the battery, a change in the state of charge ofthe battery, and the an accumulative ampere-hour throughput of thebattery during the change in the state of charge; determine, using theone or more processors, a plurality of estimated open circuit voltagesby executing the first Kalman filter based at least in part on eachbattery parameter set of the plurality of battery parameter sets;determine, using the one or more processors, a plurality of estimatedstates of charge each corresponding with one of the plurality ofestimated open circuit voltages based at least in part on the firstrelationship between the state of charge and the open circuit voltage ofthe battery; and determine, using the one or more processors, thecapacity of the battery based at least in part on execution of thesecond Kalman filter using each estimated state of charge of theplurality of estimated states of charge.
 20. The tangible,non-transitory, computer-readable medium of claim 19, comprisinginstructions to, when the battery is not relaxed at the first time andnot relaxed at the second time: determine, using the one or moreprocessors, a plurality of estimated capacities by executing the secondKalman filter using each estimated state of charge of the plurality ofestimated states of charge; determine, using the one or more processors,a mean estimated capacity of the plurality of estimated capacities;determine, using the one or more processors, an estimated capacityvariance of the plurality of estimated capacities of the battery;determine, using the one or more processors, whether a ratio of theestimated capacity variance to the mean estimated capacity squared isgreater than a relative error tolerance threshold; determine, using theone or more processors, a maximum estimated state of charge from theplurality of estimated states of charge; determine, using the one ormore processors, a minimum estimated state of charge from the pluralityof estimated state of charge; determine, using the one or moreprocessors, whether a difference between the maximum estimated state ofcharge and the minimum estimated state of charge is greater than a swingthreshold; and determine the capacity of the battery based at least inpart on the plurality of estimated capacities when the ratio of theestimated capacity variance to the mean estimated capacity squared isnot greater than the relative error tolerance threshold and thedifference between the maximum estimated state of charge and the minimumestimated state of charge is greater than the swing threshold.